Fuel supply shuts down

2003 Kia Sorento 3.5, auto, 57k miles.
After approximately 20-25 minutes of city driving ( less on the highway ), without any warning sings the fuel supply just shuts down and the car dies. After 1 minute it will start normally again and drive for couple of yards and dies again. After 15 minutes will start normally and drive for 2-3 miles and dies again.
My local garage checked and the fuel pressure was normal ( This when the engine was cold ).
Something else - for over a year the exhaust valve ( or whatever it's called ) on the fuel tank has been clogged and I can not fill gas continuously, otherwise it "spits".
On the bottom of the tank there are couple dozens very very tiny black plastic particles, maybe fro that broken valve/canister. Is it possible that they relay to the above problem, like the fuel pump sucking them up or something? Or maybe the pump is due for a replacement and if so, how come the car keeps running for so long without any hesitation before it shuts down?